Acylindrical hyperbolicity of outer automorphism groups of right-angled Artin groups

Abstract

We study the acylindrical hyperbolicity of the outer automorphism group of a right-angled Artin group $A_\Gamma$. When the defining graph $\Gamma$ has no SIL-pair (separating intersection of links), we obtain a necessary and sufficient condition for $\mathrm{Out}(A_\Gamma)$ to be acylindrically hyperbolic. As a corollary, if $\Gamma$ is a random connected graph satisfying a certain probabilistic condition, then $\mathrm{Out}(A_\Gamma)$ is not acylindrically hyperbolic with high probability. When $\Gamma$ has a maximal SIL-pair system, we derive a classification theorem for partial conjugations. Such a classification theorem allows us to show that the acylindrical hyperbolicity of $\mathrm{Out}(A_\Gamma)$ is closely related to the existence of a specific type of partial conjugations.